On this Wikipedia page there is a following sentance:
In certain cases, von Neumann stability is necessary and sufficient for stability in the sense of Lax–Richtmyer (as used in the Lax equivalence theorem): The PDE and the finite difference scheme models are linear; the PDE is constant-coefficient with periodic boundary conditions and has only two independent variables; and the scheme uses no more than two time levels.
This sentence misled me into thinking that the Von Neumann method can only be used for two time-level finite difference equations with only one spatial variable. However, I recently found this paper where the Von Neumann method was used to determine the necessary stability condition for the heat equation with two spatial variables (FTCS scheme) and I also found this paper where the Von Neumann method was used to derive the necessary stability condition for the three-time level finite difference wave equation (centered difference scheme).
Please help me understand the limitations of the Von Neumann method. My questions are:
- To how many time levels is this method restricted?
- To how many spatial variables is this method restricted?
- The examples I have seen usually are limited to second-order PDEs. Can this method be used to determine the necessary stability condition of a fourth-order PDE which has two spatial variables? Let's say the PDE is discretized by using the centered difference scheme. That means there are five time levels and five spatial levels for each spatial variable. Note that the boundary conditions are periodic and coefficients are constant. Also, the PDE is homogeneous.